An asymptotic description of vortex Kelvin modes ephane LE DIZ` By - - PDF document

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Under consideration for publication in J. Fluid Mech.

1

An asymptotic description of vortex Kelvin modes

By St´ ephane LE DIZ` ES & Laurent LACAZE

Institut de Recherche sur les Ph´ enom` enes Hors ´ Equilibre, 49, rue F. Joliot-Curie, B.P. 146, F-13384 Marseille cedex 13, France. (Received 24 September 2004)

A large-axial-wavenumber asymptotic analysis of inviscid normal modes in an axisym- metric vortex with a weak axial flow is performed in this work. Using a WKBJ approach, general conditions for the existence of regular neutral modes are obtained. Dispersion relations are derived for neutral modes confined in the vortex core (“core modes”) or in a ring (“ring modes”). Results are applied to a vortex with Gaussian vorticity and axial velocity profiles, and a good agreement with numerical results is observed for almost all values of k. The theory is also extended to deal with singular modes possessing a critical point singularity. Known damped normal modes for the Gaussian vortex without axial flow are obtained. The theory is also shown to provide explanations for a few of their peculiar properties.

• 1. Introduction

Kelvin modes are the inviscid normal modes which are associated with the rotation

• f the fluid in a stable vortex. They often describe the possible small deformations of

the vortex. They are also known to be resonantly excited in various situations (elliptic instability; precessional instability; parametric forcing). The goal of this work is to construct an asymptotic theory which provides the spatial structure and the dispersion relation of these modes. The simplest Kelvin modes are for an infinite uniform solid body rotation. In that case, there exist plane wave solutions in the rotating frame (the so-called Kelvin waves) which can be summed to form a localized inviscid normal mode (Greenspan, 1968). If the solid body rotation is within a finite cylindrical region, the frequency ω of the modes is discretized for any fixed axial wavenumber k and azimuthal wavenumber m and satisfy a dispersion relation. Moreover, in that case, Kelvin modes form a basis, so all the deformations can be expressed in terms of Kelvin modes. If the solid body rotation is limited by an irrotational fluid (Rankine vortex), the Kelvin modes satisfy similar properties (e.g. Saffman, 1992). They also form a basis for the perturbations confined within the vortex core (Arendt, Fritts & Andreassen, 1997). Kelvin modes are also known to exist, when the vorticity field is not constant. Some of their properties were analyzed for a Gaussian vortex without axial flow in Sipp & Jacquin (2003), Fabre (2002) and Fabre et al. (2004). Sipp & Jacquin (2003) used an inviscid

• approach. They showed that regular inviscid normal modes exist in a frequency interval

similar to the one obtained for the Rankine vortex; however, the interval where ω/m is in the range of the angular velocity, has to be excluded. In that frequency interval, regular inviscid normal modes do not exist anymore: they possess a critical point singularity. If this singularity is smoothed by viscosity, these modes apparently become damped with

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• S. Le Diz`

es & L. Lacaze a damping rate which is largely independent of viscosity (if sufficiently small) as shown by Fabre (2002); Fabre et al. (2004). An inviscid estimate of this damping rate can be

• btained by avoiding the singularity in the complex plane as done by Sipp & Jacquin

(2003). Such a procedure has been justified in Le Diz` es (2004) where the viscous critical layer has been resolved. In the present work, we implicitly assume a viscous problem with vanishing viscosity. This implies that, for a few modes, the path of integration of the inviscid equation has to be deformed in the complex plane, for the equation to remain asymptotically valid. In practice, this means that the critical point singularities have to be avoided in the complex plane, following the classical rule used for 2D modes in planar flows (see Lin, 1955). When an axial flow is present, regular inviscid neutral modes are still expected to exist, however very little information on their properties is available in the literature. Moreover, axial flow may promote instability in a stable vortex. For instance, the Batchelor vortex, which is a vortex with Gaussian vorticity and axial velocity profiles, is known to possess unstable inviscid modes if the axial flow is sufficiently large (see, for instance Ash & Khorrami, 1995). Here, our interest is not in these modes. Instead, we shall focus

• n vortices which are stable in a non-viscous framework. Our goal is to provide some

information on the neutral and damped modes of such vortices in a general setting using an asymptotic approach. The approach is based on a large-axial-wavenumber asymptotic analysis. In this limit, the radial structure of the normal modes varies on a faster scale than the characteristic radial scale of the base flow. These fast variations can be captured by a WKBJ theory (see for instance Bender & Orszag, 1978) and are shown to depend in a simple way

• n the base flow characteristics. For neutral modes, they are also shown to be either

pure oscillations or pure exponentials, the transition between the two types of behaviors

• ccurring at the turning points where WKBJ approximations break down.

As with the original Quantum mechanics framework, eigenmodes are constructing by forming solutions which are localized in the oscillatory regions; the dispersion relation being nothing but a discretization of the number of oscillations. In the present work, two types of modes are considered: modes confined between the vortex center and a turning point (“core modes”) and modes confined between two distant turning points (“ring modes”). The paper is organized as follows. In section 2, base flow and perturbation equations are presented. Section 3 is devoted to the large wavenumber asymptotic analysis in a general setting. Conditions for the existence of regular neutral modes in the WKBJ framework are derived. The spatial structure and the dispersion relation of core modes and ring modes are then obtained. The results are applied to a Gaussian vortex with or without axial velocity in section 4. The case without axial flow is considered first in section 4.1. In this section, the results for core modes are also extended to deal with a critical layer singularity. Both singular neutral core modes and damped core modes are obtained and compared to numerical results. In section 4.2, the asymptotic results are applied to the Gaussian vortex with axial flow (Batchelor vortex). The last section summarizes the main results and discusses a possible application of the results to the elliptic instability.

• 2. Basic flow and perturbation equations

Consider a general axisymmetric vortex with axial flow, whose velocity field may be written in cylindrical coordinates in the form : Ub(r) = (0, V (r), W(r)) . (2.1)

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An asymptotic description of vortex Kelvin modes 3 This vortex has an angular velocity Ω(r) and an axial vorticity ζ(r) given by : Ω(r) = V (r) r , (2.2a) ζ(r) = 1 r d(rV ) dr . (2.2b) In this study, viscous diffusion is not taken into account with the implicit assumption that the Reynolds number is sufficiently large. The base flow, defined by (2.1), satisfies the incompressible Euler equations regardless of the profile V and W, as long as it represents a regular field in cylindrical coordinates (in particular V (0) = 0). The asymptotic analysis detailed in the next section will be carried out for arbitrary profiles. However, in the applications, we shall only consider Gaussian vorticity and axial velocity profiles. Time and spatial scales are non-dimensionalized by the angular velocity in the vortex center, and the core size, respectively; such that Ω(r) and W(r) read : Ω(r) = 1 − e−r2 r2 , (2.3a) W(r) = W0e−r2 , (2.3b) where W0 is a constant measuring the strength of the axial flow. We shall be concerned with inviscid linear perturbations in the form of normal modes : (U, P) = (u, v, w, p)eikz+imθ−iωt , (2.4) where k and m are axial and azimuthal wavenumbers and ω is the frequency. The equations for the velocity and pressure amplitudes (u, v, w, p) are : iΦu − 2Ωv = −dp dr (2.5a) iΦv + ζu = −imp r (2.5b) iΦw + W ′w = −ikp (2.5c) 1 r d(ru) dr + imv r + ikw = 0 , (2.5d) where a prime denotes a derivative with respect to r, and Φ(r) = −ω + mΩ(r) + kW(r) . (2.6) Equations (2.5a-d) can be reduced to a single equation for the pressure p (see Saffman, 1992; Le Diz` es, 2004) to form : d2p dr2 + 1 r − ∆′ ∆ dp dr + 2m rΦ∆(Ω′∆ − Ω∆′) + k2∆ Φ2 − m2 r2 − 2mkW ′Ω rΦ2

• p = 0 ,

(2.7) where ∆(r) = 2ζ(r)Ω(r) − Φ2(r) . (2.8) If ∆ and Φ do not vanish at zero, the condition that p remains bounded at ∞ and at r = 0 transforms equation (2.7) into an eigenvalue problem for ω (assuming k and m are fixed). The case where Φ(0) is close to zero will not be considered here. It requires a specific study by itself. We refer to Fabre (2002) for the Gaussian vortex without axial

• flow. Partial results for the Batchelor vortex can also be found in Stewartson & Leibovich

(1987) and Stewartson & Brown (1985). The objective of this work is to provide information on the dispersion relation and on

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• S. Le Diz`

es & L. Lacaze the spatial structure of the eigenmode. Our approach is based on an asymptotic analysis for large k.

• 3. Large k asymptotic analysis

In this section, the asymptotic analysis is presented in a general framework. Applica- tions are considered in the next section. The principle of the analysis is to construct approximate solutions valid in the limit k → ∞. For large k, when there is no axial flow, or if the axial flow scales as 1/k, the expression before p in equation (2.7) becomes particularly simple as it reduces to a single term k2∆/Φ2. Therefore, for large k, this term has to be equilibrated by rapid variation

• f the pressure amplitude on the scale rk. Such variations can be captured by a WKBJ

analysis (see Bender & Orszag, 1978). In this framework, the perturbation pressure is expanded as p =

• p0(r) + p1(r)

k + · · ·

• ekφ(r) .

(3.1) The expression for φ(r) is obtained at the order k2: dφ dr 2 = − ∆ Φ2 , (3.2) where we have assumed in the expression (2.6) for Φ that the axial flow is small and can be written as kW ≡ W1 = O(1) . (3.3) From equation (3.2), it follows that : φ(r) = ±i r √ ∆ Φ dr . (3.4) At the order k, an equation for p0(r) is obtained : 2φ′ dp0 dr +

• φ′

1 r − ∆′ ∆

• + φ′′
• p0 = 0 ,

(3.5) which gives, for both functions φ, p0(r) =

• Φ

r ∆1/4 . (3.6) Expressions (3.1), (3.4) and (3.6) provide two independent leading order approxima- tions of solutions to (2.7). These so-called WKBJ approximations break down at the vortex center r = 0, and at the points where Φ or ∆ vanishes. The vortex center is a regular singularity which comes from the use of cylindrical coordinates. As shown below, this singularity can be easily smoothed by carrying out a local analysis for r = O(1/k). Points where ∆ = 0 are the so-called turning points of the WKBJ approximations. In the neighborhood of these turning points, the two approximations are no longer inde-

• pendent. One can also show that higher order corrections, such as p1 in the expansion

(3.1), diverge at turning points. These turning point singularities can also be resolved by a local analysis of the turning point region (Bender & Orszag, 1978, see also below). Finally, the singularities where Φ = 0, i. e. ω = mΩ+W1, are the so-called critical points

• f the inviscid approximation. As our choice is to stay inviscid, we shall not resolve these

singularities here. Instead, if such a singularity appears, it will be avoided by deform- ing the integration space of equation (2.7) in the complex r-plane, in order to stay in

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An asymptotic description of vortex Kelvin modes 5 the regions of the complex plane where the inviscid approximation remains valid (Sipp & Jacquin, 2003; Le Diz` es, 2004). In those cases, the inviscid solutions would become singular in the physical domain. If we restrict for a moment our attention to regular neutral eigenmodes, a few results can be obtained in a general setting. By definition, for those modes, both the frequency ω and the wavenumber k are real and Φ(r) never vanishes on the real axis. The WKBJ approximations constructed for the present problem are then very similar to those initially introduced by Wentzel, Kramers and Brillouin for describing the bounded states of a particule in a potential well in Quantum Mechanics(see Landau & Lifchitz, 1966). If ∆ > 0, WKBJ approximations are oscillating functions, if ∆ < 0, they are exponentials. In the semi-classical description of Quantum Mechanics, this corresponds to oscillating wave functions in regions where the energy level is larger than the local potential and evanescent exponentials where it is smaller. As in this framework where it is proved than there is no energy level smaller than the potential minimum, one can prove here that there does not exist regular neutral eigenmode for which ∆ remains positive for all r. Indeed if ∆ > 0 for all r, both WKBJ approximations are uniformly valid in any interval of ]0, +∞[, and no combination of these approximations can be matched to solutions which are bounded at the origin and at infinity (see also below). The conclusion is therefore that ∆ must be non-negative somewhere for a regular neutral mode to exist. To analyse this condition of existence, it is useful to define what is often called the epicyclic frequencies ω±(r) of the vortex at the radial coordinate r : ω± = mΩ(r) + W1(r) ±

• 2Ω(r)ζ(r) .

(3.7) In this expression, the quantity Υ(r) = 2Ω(r)ζ(r) is what is called the Rayleigh dis- criminant. It characterizes the unstable character of the vortex with respect to the centrifugal instability (see Drazin & Reid, 1981). In the stable vortex we consider, Υ is always non-negative which implies that ω+ and ω− are real functions. These two func- tions provide the frequency interval where ∆ is positive, that is ∆(r) > 0 if and only if ω−(r) < ω < ω+(r). It is also useful to consider the function ωc(r) = mΩ(r) + W1(r) , (3.8) which provides the (critical) frequency of the mode that exhibits a critical point at the radial location r. One can now easily deduce the frequency intervals where regular neutral modes can exist. Their frequency must be somewhere between ω− and ω+ without being in the range of ωc. The regular neutral mode frequencies then satisfy min(ω−) ≤ ω ≤ min(ωc) , (3.9)

• r

max(ωc) ≤ ω ≤ max(ω+) . (3.10) Moreover, the upper bound in (3.9) and the lower bound in (3.10) can be excluded if the extrema are reached for finite r. In the Quantum Mechanics framework, bounded states are known to be discretized by their number of oscillations in the potential well (see Landau & Lifchitz, 1966). We shall see below that the same result is obtained here: eigenmodes will be localized in the region where ∆ > 0 and selected by a discretization condition on their number of

• scillations in that region. In the rest of this section, we shall obtain this discretization

condition when there is a single interval of positive ∆. More precisely, we shall assume that the functions ∆and Φ satisfy one of the two hypotheses:

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• S. Le Diz`

es & L. Lacaze Hypothesis H1: The function ∆ is positive for 0 ≤ r < rt, negative for r > rt and has a single zero rt. The function Φ does not vanish on the real axis. Hypothesis H2: The function ∆ is positive for r1 < r < r2, negative for 0 ≤ r < r1 and r2 < r, and has two simple zeroes r1 and r2. The function Φ does not vanish on the real axis. When Hypothesis H1 is satisfied, eigenmodes are localized between 0 and rt. We shall denote such modes as “core modes”. When Hypothesis H2 is satisfied, eigenmodes are localized between r1 and r2 and for this reason are termed “ring modes”. For the vortices considered in section 4, regular neutral modes will be found to be either core modes or ring modes. However, for vortices with a more complex profile, one could imagine more complex modes, corresponding to configurations with multiple distinct regions where ∆ is positive. Each type of mode would require a specific analysis, but it can follow the approach which is now presented for “core modes” and “ring modes”. In section 4, it will also be shown how the above hypotheses can be extended to deal with complex frequencies. 3.1. Core modes When Hypothesis H1 is satisfied, the mode structure can be decomposed into four re- gions. (1) The neighborhood of the center r = 0. (2) The “core” region between 0 and rt. (3) The neighborhood of the turning point rt. (4) The “outer” region for r > rt. In each region, a specific approximation of the mode is obtained. The condition of match- ing of the different approximations will provide the dispersion relation. The neighborhood of r = 0. In order to smooth the singularity of the WKBJ approximations at r = 0 we introduce the local variable ¯ r = kr and expand the perturbation pressure as ¯ p(¯ r) = ¯ p0(¯ r) + ¯ p1(¯ r) k + · · · . (3.11) At leading order, ¯ p0 is found to satisfy : d2¯ p0 d¯ r2 + 1 ¯ r d¯ p0 d¯ r + ∆(0) Φ2(0) − m2 ¯ r2

• ¯

p0 = 0 . (3.12) The solution which is bounded at ¯ r = 0 is given by : ¯ p0 = a0J|m|(β0¯ r) , (3.13) where a0 is a constant, J|m| is the usual Bessel function of first kind and β0 is a positive constant given by : β0 =

• ∆(0)

Φ(0) . (3.14) The “core” region (0 < r < rt) In the “core” region, the WKBJ approximations are valid and are oscillating functions.

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An asymptotic description of vortex Kelvin modes 7 The matching with the solution valid in the neighborhood of r = 0 provides a condition

• n the solution in this region.

The function Jm(z) in (3.13) has the following expansion (see Abramowitz & Stegun, 1965) for large |z| : J|m|(z) ∼

• 2

πz cos

• z − |m|π

2 − π 4

• ,

|arg(z)| < π . (3.15) This guarantees that the leading order expression (3.13) can match (as ¯ r → ∞) a com- bination of WKBJ approximations : p ∼ A+p0(r)ekφ + A−p0(r)e−kφ (3.16) provided that, A+ekφ(0) = a0

• 2π∆(0)k

e−i|m|π/2−iπ/4 , (3.17a) A−e−kφ(0) = a0

• 2π∆(0)k

ei|m|π/2+iπ/4 , (3.17b) that is, A± = A0e∓kφ(0) exp

• ∓i|m|π

2 ∓ iπ 4

• .

(3.18) It follows that a leading order approximation for the solution in this region is given by p ∼ A0

• Φ

r ∆1/4 cos

• k

r √ ∆ Φ dr − |m|π 2 − π 4

• ,

(3.19) where A0 is a constant which can be expressed in term of a0. The “outer” region (r > rt) In the “outer” region, one of the WKBJ approximations is exponentially increasing while the other is exponentially decreasing. In order to form a solution which vanishes for large r, the exponentially growing WKBJ approximation should not be present in the

• solution. It follows that for r > rt, the solution can be written at leading order as :

p ∼ B0

• Φ

r (−∆)1/4 exp

• −k

r

rt

√ −∆ Φ

• dr .

(3.20) The matching of the “outer” region with the “core” region is performed in the neigh- borhood of the turning point rt. This provides the dispersion relation and a relation between the coefficients A0 and B0 of expressions (3.19) and (3.20). Neighborhood of the turning point rt The local analysis of the neighborhood of a simple turning point is classical (see for instance Bender & Orszag, 1978). Following the textbooks, one introduces a local variable ˜ r = (r − rt)k2/3, where the power 2/3 is typical of a simple turning point analysis, and expands the perturbation pressure as : ˜ p(˜ r) = ˜ p0(˜ r) + k−1/3˜ p1(˜ r) + · · · (3.21) At leading order, an equation is obtained for ˜ p0 : d2˜ p0 d˜ r2 − 1 ˜ r d˜ p0 d˜ r + ∆′

t˜

r Φ2

t

˜ p0 = 0 . (3.22)

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• S. Le Diz`

es & L. Lacaze This equation can be integrated as : ˜ p0(˜ r) = b0A′

i(κ˜

r) + c0B′

i(κ˜

r) , (3.23) where b0 and c0 are constants, Ai(z) and Bi(z) are Airy functions (see Abramowitz & Stegun, 1965) and κ = (−∆′

t/Φ2 t)1/3.

Dispersion relation The matching of the “turning point region” with the “outer” region requires that the exponentially growing function B′

i in (3.23) should not be present in the solution, that is

c0 = 0. Using the following expansions (Abramowitz & Stegun, 1965) of Ai′(z) for large |z| : Ai′(z) ∼ − 1 2√π z1/4 exp

• −2

3z3/2

• ,

|arg(z)| < π , (3.24a) Ai′(−z) ∼ − 1 √π z1/4 cos 2 3z3/2 + π 4

• ,

|arg(z)| < 2 3π , (3.24b) we obtain the relation : − b0 2√π κ1/4k1/6 = B0

• Φt

rt (−∆′

t)1/4 ,

(3.25) from the matching with the “outer” region, and − b0 √π κ1/4k1/6e−iπ/4 = A0

• Φt

rt (−∆′

t)1/4 exp

• ik

rt √ ∆ Φ dr − i|m|π 2 − iπ 4

• ,

(3.26a) − b0 √π κ1/4k1/6eiπ/4 = A0

• Φt

rt (−∆′

t)1/4 exp

• −ik

rt √ ∆ Φ dr + i|m|π 2 + iπ 4

• ,

(3.26b) from the matching with the “core” region. Equations (3.25) and (3.26) yield A2

0 = 4B2

(3.27) and the dispersion relation that links k, m and ω: exp

• 2ik

rt √ ∆ Φ dr − i|m|π

• = 1 ,

(3.28) which can also be written as : k = |m|π + 2nπ 2 rt

√ ∆ Φ

dr , where n is a non-negative integer. (3.29) We recall that in the above expression, Φ and ∆ are given by expressions (2.6)and (2.8) respectively and that rt is a zero of ∆. Expression (3.29) is the dispersion relation for “core” modes in the limit of large k. Spatial structure of the eigenmodes Approximations for the pressure perturbation can now be obtained in each region using expressions (3.25) and (3.27). They depend on a unique amplitude factor A0 which can be fixed to 1 such that a0, B0 and b0 are now given by a0 =

• π∆0k

2 (3.30a)

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An asymptotic description of vortex Kelvin modes 9 B0 = (−1)n 2 (3.30b) b0 = −k−1/6(−1)n π rt Φ2/3

t

(−∆′

t)1/6 .

(3.30c) Approximations for the velocity field are also easily derived from p using the system (2.5) which gives u = −iΦ ∆ dp dr − 2imΩ r∆ p (3.31a) v = ζ ∆ dp dr + mΦ r∆ p (3.31b) w = − k Φp . (3.31c) We obtain the following expressions. In the “core” region: p ∼

• Φ

r ∆1/4 cos

• k

r √ ∆ Φ dr − |m|π 2 − π 4

• ,

(3.32a) u ∼ ik

• Φ

r ∆−1/4 sin

• k

r √ ∆ Φ dr − |m|π 2 − π 4

• ,

(3.32b) v ∼ −kζ∆−1/4 √ rΦ sin

• k

r √ ∆ Φ dr − |m|π 2 − π 4

• ,

(3.32c) w ∼ −k∆1/4 √ rΦ cos

• k

r √ ∆ Φ dr − |m|π 2 − π 4

• .

(3.32d) In the neighborhood of rt (r − rt = O(k−2/3)): p ∼ −k−1/6(−1)n π rt Φ2/3

t

(−∆′

t)1/6A′ i

• κ(r − rt)k2/3

(3.33a) u ∼ −ik7/6(−1)n π rt Φ1/3

t

(−∆′

t)−1/6Ai

• κ(r − rt)k2/3

(3.33b) v ∼ k7/6(−1)n π rt Φ−2/3

t

(−∆′

t)−1/6ζtAi

• κ(r − rt)k2/3

(3.33c) w ∼ k5/6(−1)n π rt Φ−1/3

t

(−∆′

t)1/6A′ i

• κ(r − rt)k2/3

(3.33d) with κ = (−∆′

t/Φ2 t)1/3.

Near the origin r = O(1/k): p ∼

• π∆0k

2 J|m|(β0kr) (3.34a) u ∼ −i π 2 k3/2

• J′

|m|(β0kr) + 2mΩ0

√∆0krJ|m|(β0kr)

• (3.34b)

v ∼ π 2 k3/2 ζ0 Φ0 J′

|m|(β0kr) −

mΦ0 √∆0krJ|m|(β0kr)

• (3.34c)

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• S. Le Diz`

es & L. Lacaze w ∼ −

• π∆0

2Φ2 k3/2J|m|(β0kr) (3.34d) with β0 = √∆0/Φ0. In the “outer” region: p ∼ (−1)n 2

• Φ

r (−∆)1/4 exp

• −k

r

rt

√ −∆ Φ dr

• ,

(3.35a) u ∼ −ik (−1)n 2

• Φ

r (−∆)−1/4 exp

• −k

r

rt

√ −∆ Φ dr

• ,

(3.35b) v ∼ (−1)n 2 k(−∆)−1/4 √ rΦ exp

• −k

r

rt

√ −∆ Φ dr

• ,

(3.35c) w ∼ −(−1)n 2 k(−∆)1/4 √ rΦ exp

• −k

r

rt

√ −∆ Φ dr

• .

(3.35d) These expressions will be compared to numerical results in the applications considered in section 4. 3.2. Ring modes When hypothesis H2 is satisfied, the eigenmode structure can be divided in the following regions: (1) The neighborhood of the origin. (2) The “outer” region I for 0 < r < r1. (3) The neighborhood of the turning point r1. (4) The “ring” region for r1 < r < r2. (5) The neighborhood of the turning point r2. (6) The “outer” region II for r > r2. The analysis is very similar to the one performed for “core modes”. In the neighbor- hood of the origin, the pressure is expressed in terms of Bessel functions. A leading

• rder approximation is given by (3.13) but β0 is now purely imaginary. Near the origin,

the pressure can be written as : ¯ p0 = a0I|m|(γ0¯ r) , (3.36) where a0 is a constant, I|m| is the usual Bessel function of second kind and, γ0 =

• −∆(0)

Φ(0) . (3.37) For large ¯ r, expression (3.36) becomes exponentially large. In the outer region I, the solution can therefore be approximated by a single WKBJ wave : p ∼ AI

• Φ

r (−∆)1/4 exp

• k

r √ −∆ Φ dr

• ,

(3.38) where the matching imposes a relation between AI and a0. Note that if ∆ was negative everywhere, the Outer region would extend up to infinity, and the approximation (3.38) would be unbounded for large r, invalidating the boundary condition at +∞. This justifies the condition of existence stated above which requires a region of positive ∆. In the outer region II, the solution is, as above, given by the subdominant WKBJ

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An asymptotic description of vortex Kelvin modes 11 approximation : p ∼ AII

• Φ

r (−∆)1/4 exp

• −k

r

r2

√ −∆ Φ dr

• .

(3.39) The solution in the turning point region near r2 which matches this expression is, as above : p ∼ a2A′

i(κ2ˇ

r) , (3.40) where ˇ r = (r − r2)k2/3 and κ = (−∆′

2/Φ2 2)1/3.

Similarly, the solution in the turning point region near r1 which matches expression (3.38) is : p ∼ a1A′

i(κ1˜

r) , (3.41) where ˜ r = (r − r1)k2/3 and κ = −(∆′

1/Φ2 1)1/3.

Expressions (3.41) and (3.40) imply that in the “ring” region, the solution admits approximations of the form : p ∼ A0

• Φ

r ∆1/4 cos

• k

r

r1

√ ∆ Φ dr + π/4

• ,

(3.42) and p ∼ B0

• Φ

r ∆1/4 cos

• k

r

r2

√ ∆ Φ dr − π/4

• ,

(3.43) where A0 and B0 can be expressed in terms of a1 and a2 respectively. These two expres- sions are compatible only if sin

• k

r2

r1

√ ∆ Φ dr + π/2

• = 0 ,

that is, k = nπ + π/2 r2

r1 √ ∆ Φ

, where n is a non-negative integer. (3.44) Expression (3.44) is the dispersion relation for “ring modes” in the limit of large k. Spatial structure of the eigenmodes As for the core modes, approximations for the pressure and the velocity field of ring modes can now easily be obtained in each region. If we fix A0 = 1, we get in each region the following expressions: In the “ring” region p ∼

• Φ

r ∆1/4 cos

• k

r

r1

√ ∆ Φ dr + π 4

• ,

(3.45a) u ∼ ik

• Φ

r ∆−1/4 sin

• k

r

r1

√ ∆ Φ dr + π 4

• ,

(3.45b) v ∼ −kζ∆−1/4 √ rΦ sin

• k

r

r1

√ ∆ Φ dr + π 4

• ,

(3.45c) w ∼ −k∆1/4 √ rΦ cos

• k

r

r1

√ ∆ Φ dr + π 4

• .

(3.45d)

SLIDE 12

12

• S. Le Diz`

es & L. Lacaze In the region near the turning point r1, defined by r − r1 = O(k−2/3): p ∼ k−1/6√πΦ2/3

1

r−1/2

1

(∆′

1)1/6A′ i

• κ1(r − r1)k2/3

, (3.46a) u ∼ −ik7/6√πΦ1/3

1

r−1/2

1

(∆′

1)−1/6Ai

• κ1(r − r1)k2/3

, (3.46b) v ∼ k7/6√πΦ−2/3

1

r−1/2

1

(∆′

1)−1/6ζ1Ai

• κ1(r − r1)k2/3

, (3.46c) w ∼ −k5/6√πΦ−1/3

1

r−1/2

1

(∆′

1)1/6A′ i

• κ1(r − r1)k2/3

, (3.46d) with κ1 = (∆′

1/Φ2 1)1/3.

In the region near the turning point r2, defined by r − r2 = O(k−2/3): p ∼ −k−1/6(−1)n√πΦ2/3

2

r−1/2

2

(−∆′

2)1/6A′ i

• κ2(r − r2)k2/3

, (3.47a) u ∼ −ik7/6(−1)n√πΦ1/3

2

r−1/2

2

(−∆′

2)−1/6Ai

• κ2(r − r2)k2/3

, (3.47b) v ∼ k7/6(−1)n√πΦ−2/3

2

r−1/2

2

(−∆′

2)−1/6ζ2Ai

• κ2(r − r2)k2/3

, (3.47c) w ∼ k5/6(−1)n√πΦ−1/3

2

r−1/2

2

(−∆′

2)1/6A′ i

• κ2(r − r2)k2/3

. (3.47d) with κ2 = (−∆′

2/Φ2 2)1/3.

In the “outer” region I: p ∼

• Φ

4r(−∆)1/4 exp

• k

r

r1

√ −∆ Φ dr

• ,

(3.48a) u ∼ ik

• Φ

4r(−∆)−1/4 exp

• k

r

r1

√ −∆ Φ dr

• ,

(3.48b) v ∼ −k(−∆)−1/4 2 √ rΦ exp

• k

r

r1

√ −∆ Φ dr

• ,

(3.48c) w ∼ −k(−∆)1/4 2 √ rΦ exp

• k

r

r1

√ −∆ Φ dr

• ,

(3.48d) In the “outer” region II: p ∼ (−1)n 2

• Φ

r (−∆)1/4 exp

• −k

r

r2

√ −∆ Φ dr

• ,

(3.49a) u ∼ −ik (−1)n 2

• Φ

r (−∆)−1/4 exp

• −k

r

r2

√ −∆ Φ dr

• ,

(3.49b) v ∼ (−1)n 2 k(−∆)−1/4 √ rΦ exp

• −k

r

r2

√ −∆ Φ dr

• ,

(3.49c) w ∼ −(−1)n 2 k(−∆)1/4 √ rΦ exp

• −k

r

r2

√ −∆ Φ dr

• ,

(3.49d) The above approximations for ring modes will be compared to numerical solutions in section 4.2.

SLIDE 13

An asymptotic description of vortex Kelvin modes 13

1 2 3 4 5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 r ω +,ω −,ωc (a) 1 2 3 4 5 −1 −0.5 0.5 1 1.5 2 2.5 3 r ω +,ω −,ω0 (b) 1 2 3 4 5 0.5 1 1.5 2 2.5 3 3.5 4 r ω +,ω −,ω0 (c) 1 2 3 4 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 r ω +,ω −,ω0 (d)

Figure 1. The functions ω± (solid lines) and ωc (dashed line) versus r for the Lamb vortex. (a): m = 0; (b): m = 1; (c): m = 2; (d): m = 3.

• 4. Applications

4.1. The Gaussian vortex without axial flow (Lamb vortex) In this section, we consider a Gaussian vortex without axial flow. The base flow profile is given by (2.3) with W0 = 0. In this case, the functions ω± and ωc have a limited number of possible behaviors. In figures 1(a-d) the functions ω± and ωc are plotted for m = 0, 1, 2, 3. For larger values of m (m > 3), results are similar to figure 1(d): the three functions ω+, ω− and ωc are monotonically decreasing to zero; their values at r = 0 are ω+(0) = m + 2, ω−(0) = m − 2 and ωc(0) = m. The functions ω± and ωc, for negative m, are obtained by making the transformations m → −m and ω → −ω. 4.1.1. Regular neutral core modes The conditions (3.9) and (3.10) for the existence of regular neutral modes give here −2 ≤ ω ≤ 2 for m = 0, −1 ≤ ω ≤ 0 and 1 < ω ≤ 3 for m = 1, and m < ω ≤ m + 2 for m ≥ 2. Inspections of figures 1(a-d) shows that in all these frequency intervals, Hypothesis H1 is satisfied. We therefore expect all regular neutral modes of the Lamb vortex to be core modes. In these frequency intervals, formula (3.29) can be applied. The results are displayed in solid lines in figures 2(a-d) for the first branches (n = 1, 2, 3, 4). In the same figures, dotted lines represent the dispersion relation obtained by a numerical integration of equation (2.7). These figures demonstrate the good agreement of the large- k dispersion relation with the numerics for not only large k, but also for small values of

• k. The asymptotic results also tend to be better for small m. For m = 0, the asymptotic

predictions are almost identical to the numerical results for all values of k.

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2 4 6 8 10 −2 −1.5 −1 −0.5 0.5 1 1.5 2 k ω (a) 2 4 6 8 10 −1 −0.5 0.5 1 1.5 2 2.5 k ω (b) 2 4 6 8 10 2 2.2 2.4 2.6 2.8 3 3.2 k ω (c) 2 4 6 8 10 3 3.2 3.4 3.6 3.8 4 4.2 k ω (d)

Figure 2. Dispersion relation of neutral core modes of the Lamb vortex. Numerical results (dotted lines) and large-k asymptotic results [expression (3.29) for n = 1, 2, 3, 4] (solid lines). (a): m = 0; (b): m = 1; (c): m = 2. (d): m = 3. The dashed line in figure (b) is expression (3.29) with n = 0.

For m = 1, note that there is an additional branch for ω < 0 in the numerics. This branch turns out to be associated with the n = 0 mode in expression (3.29). For large k, a good agreement is indeed obtained as demonstrated in figure 2(b). It is worth mentioning that the n = 0 mode does not exist for ω > 1. For all m ≥ 1, the branches associated with frequencies in the interval m < ω ≤ m + 2 (that is, all the branches if m ≥ 2) satisfy the same property. Their frequency starts from ω = m at k = 0, and grows monotonically with k, to ω = m + 2. The vanishing

• f k as ω → m+ is due to the displacement of a critical point toward the origin which

makes the integral in expression (3.29) divergent. As explained above, the results for negative m are obtained by making the transfor- mations m → −m and ω → −ω. 4.1.2. Singular neutral core modes For m = 1, the neutral branches, obtained in the previous section, stop abruptly when ω reaches 0. For small positive frequencies, a critical point singularity rc is now present and ∆ changes sign near this point (as seen on figure 1(b)). This invalidates hypothesis

• H1. If we wanted to stay on the real axis, near such a critical point, viscous effects would

have to be reintroduced to smooth the singularity. As mentioned above, the inviscid equation can however remain valid if we avoid the critical point in the complex plane following the rule of contour deformation prescribed by Lin (1955). This rule, which can

SLIDE 15

An asymptotic description of vortex Kelvin modes 15 be justified in the present context by using Le Diz` es (2004) results, states that the side where the contour is deformed, is obtained by considering the displacement of the critical point for weakly amplified frequencies: if the critical point goes in the lower quadrants (ℑm(rc) < 0), one has to deform the contour above the critical point, if it goes in the upper quadrants, one has to deform the contour below. In the following, this rule is systematically applied. The displacement of the critical points is monitored and it is always checked that the integration contour remains in the region of the complex plane, where the inviscid equation is asymptotically valid. It is worth mentioning that by this procedure, the inviscid limit of a viscous eigenvalue is obtained but the corresponding eigenmode is nolonger regular on the real axis. On the real axis, the eigenmode is expected to exhibit viscous oscillations which are not described by the present framework (see Fabre et al., 2004). The deformation of the contour in the complex plane also implies constraints on the large k analysis. Indeed, if the critical point shifts into the complex-r plane the validity of the WKBJ approximations in the complex plane has to be considered. In principle, this requires the analysis of the characteristic curves associated with the WKBJ approxima- tions defined as ℜe(φ) = Constant and ℑm(φ) = Constant. Among these curves, Stokes lines and Anti-Stokes lines are known to play a particular role (Olver, 1997; Fedoryuk, 1993); they are defined respectively by : ℜe r

rt

√ −∆ Φ dr = 0 , (4.1) and ℑm r

rt

√ −∆ Φ dr = 0 , (4.2) where rt is any turning point. In the present work, we mostly use the following result which can be deduced from the theorem 11.1 of Olver (1997): The WKBJ approximations are uniformly valid on any sufficiently regular finite path along which φ and p0 are holomorphic and ℜe(φ) is monotonic. Using Olver’s terminology, we shall designate such a path as a progressive path. Note in particular that a part of a Stokes line which does not contain turning points and critical points is a progressive path. In this section and in the following section, our objective is to demonstrate that the asymptotic analysis

• f section 3.1 also applies to complex-plane configurations. The main argument of the

analysis is based on the fact that the matching procedure may be performed by the same method provided that we remain on progressive paths. This guarantees that the WKBJ approximations remain uniformly valid in each region. The boundary conditions at infinity and at the origin are then transmitted up to the turning point region without

• modification. For the core modes, the matching at rc can then be performed in a similar

way and it leads to the same dispersion relation. For small positive values of ω, the Stokes line structure around the real axis has the typical form displayed in figure 3. The critical point indicated by a star is close to two additional turning points indicated by black circles. What is remarkable is that one can find a progressive path which connects the first turning point to infinity by avoiding the critical point and the two nearby turning points in a complex region where the inviscid equation is expected to remain valid. The progressive character of the path is shown by checking that ℜe(φ), where φ is given by (3.4), is a monotonic function along the

• path. This check has been performed and the results are shown in figure 4 for the path

indicated by a dotted line in figure 3. As WKBJ approximations are uniformly valid along this progressive path, it can re-

SLIDE 16

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• S. Le Diz`

es & L. Lacaze

0.5 1 1.5 2 2.5 3 3.5 4 −0.2 −0.15 −0.1 −0.05 0.05 0.1 0.15 0.2 real(r) imag(r)

Figure 3. Stokes lines (solid lines) and anti-Stokes lines (dashed lines) in the complex r plane for the Lamb vortex and ω = 0.12 and m = 1. Black circles are turning points, the star is a critical point. The dotted curve represents a progressive path along which the integration can be carried out.

0.5 1 1.5 2 2.5 3 −3 −2.5 −2 −1.5 −1 −0.5 0.5 s real(φ)

Figure 4. Evolution of the function ℜe(φ) = ℜe( r(s)

rt

√ ∆/Φdr) along the path s → r(s) indicated by a dotted line in figure 3.

place the “real outer region” which was considered in the previous section. The core region is as previously the interval ]0, rt[. As this interval is along a Stokes line, it is also a progressive path. Finally the matching conditions across rt can now be applied as in section 3.1, if the progressive path associated with the outer region reaches rt on the op- posite side of rt with respect to the core region. This condition can be expressed in term

• f Stokes lines: the “outer” progressive path must be in the (Stokes) sector delimited

by the 2 other Stokes lines issued from rt (i.e. different from the Stokes line associated with the core region). The matching leads to the dispersion relation (3.29) where rt is the first (smallest) turning point. The typical structure (shown in figure 3), which leads to this result, is obtained in the following cases :

• m = 1: 0 < ω < ω(1)

c

≈ 0.1267

• m = 2: 0 < ω < ω(2)

c

≈ 0.3871 The critical frequencies ω(1)

c

and ω(2)

c

are the frequencies for which the first and second

SLIDE 17

An asymptotic description of vortex Kelvin modes 17 turning points collide for m = 1 and m = 2, respectively. These frequencies are also visible in figures 1(b,c), they correspond to the maximum values of ω−(r). If one applies relation (3.29) in these frequency intervals, one obtains the branches which are plotted in thick solid lines in figure 5(a) for m = 1 and 7(a) for m = 2. As will be discussed in more detail below, the agreement with numerical results (dotted lines) is very good. However, it is noteworthy that the numerical frequencies possess a small negative imaginary part when the first and second turning points are close to each other. This is visible in figure 5(b) for m = 1 close to ℜe(ω) ≈ 0.12, and in figure7(b) for m = 2 close to ℜe(ω) ≈ 0.35. We think that this damping effect could be associated with higher

• rder corrections in 1/k in the asymptotic analysis.

The singular neutral modes described here do not exist for m ≥ 3 when no axial flow is present. We shall see below however that they can appear for other values of m when an axial flow is added. 4.1.3. Singular damped core modes In this section, we demonstrate that formula (3.29) can also be applied to obtain damped modes. The principle has been explained above. It is to replace the real intervals by complex progressive paths. The core region between 0 and rt is now a complex progressive path along which the two WKBJ approximations are oscillatory. This means that the core region is along a Stokes line connecting rt to the vortex center. The turning point region is in the complex plane, and the outer region is a complex progressive path that goes from rt to +∞ (infinity on the real axis) and which leaves rt on the opposite side of the core region, as explained above. In addition, the whole path that goes from 0 to ∞ (Stokes line between 0 and rt and progressive path between rt and +∞) must avoid the critical point singularity as prescribed by Lin’s rule. Checking these conditions requires a fine analysis of the Stokes lines network and a monitoring of the evolution of turning points and critical points as the parameters are varied. Indeed, there are several turning points in the complex plane, so one has to check that an appropriate choice is made in expression (3.29). Note that by contrast with the neutral modes, the integral in expression (3.29) has to be calculated in the complex plane, as rt is now a complex number. Examples of Stokes line structures obtained for damped eigenfrequencies are shown in figure 6(a,b) for m = 1. The results for this value of m are shown in figures 5(a,b). Figure 5(a) displays the real part of the frequency versus k for the first four branches. Figure 5(b) shows ℜe(ω) versus ℑm(ω). Formula (3.29) tells us that, for all the branches, the frequencies should be on the same curve. This curve is given by the vanishing of the imaginary part in (3.29), i. e. ℑm rt √ ∆ Φ dr

• = 0 .

(4.3) This condition is the condition mentioned above, that is one of the Stokes lines leaving the turning point rt should pass through the origin. In figure 5(a,b) the numerical results obtained by Sipp & Jacquin (2003) by a non- viscous shooting method with contour deformation are shown in dotted lines. As can be seen, the agreement with the theory is very good. Of note is the convergence of all branches as k goes to zero to a single frequency ω ≈ 0.0474 − 0.1144i. As pointed out by Sipp & Jacquin (2003), at this frequency, the integration contour is pinched between two critical points. This is clearly visible on the Stokes line network shown in figure 6(b). In

• ur case, this means that a singularity (which cannot be removed) appears in the integral

SLIDE 18

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• S. Le Diz`

es & L. Lacaze

1 2 3 4 5 6 7 8 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 k real(ω) (a) −0.16 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 imag(ω) real(ω) (b)

Figure 5. Dispersion relation of singular core modes of the Lamb vortex for m = 1. (a) ℜe(ω) versus k. (b) ℜe(ω) versus ℑm(ω). The thick solid lines indicate the neutral modes obtained in section 4.1.2. The dotted lines are inviscid numerical results by Sipp & Jacquin (2003). The Stokes line structures of the eigenfrequencies indicated by stars in figure (b) are displayed in figure 6.

0.5 1 1.5 2 2.5 3 3.5 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 real(r) imag(r) 0.5 1 1.5 2 2.5 3 3.5 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 real(r) imag(r)

(a) (b) Figure 6. Stokes line networks for m = 1 modes of the Lamb vortex. (a) ω = 0.095 − 0.142i; (b) ω = 0.05−0.119i. Stars are critical points and black circles are turning points. The relevant Stokes line network is indicated by a thick solid line. Short lines indicate on a grid the direction

• f the characteristic curves ℜe(φ) = Constant.
• f formula (3.29). The consequence is that the integral becomes divergent, which implies

that k has to go to zero. This behavior is typical: as soon as the integration contour is pinched between two critical points, the wavenumber decreases to zero. In figures 7(a,b) are the results for m = 2 together with numerical results by Fabre et al. (2004). The results by Fabre et al. (2004) have been obtained by an inviscid spectral method. As in Sipp & Jacquin, the integration contour has been deformed in the complex plane. Fabre et al. (2004) have also considered viscous effects. They have demonstrated that both m = 1 and m = 2 damped modes could indeed be obtained by a viscous code with small viscosity, without deforming the integration contour. This confirms that a correct integration contour has been chosen. What is surprising, in figure 7(a), is the discontinuous behavior of the spatial branches. This discontinuity corresponds to the branch bifurcation shown in figure 7(b) at ω ≈ 0.1285 − 0.307i. This strange behavior can be traced back to a topological change of the Stokes line structures. The Stokes line network at the crossing point frequency is shown in figure 8(b). Before

SLIDE 19

An asymptotic description of vortex Kelvin modes 19

2 4 6 8 10 12 −0.1 0.1 0.2 0.3 0.4 k real(ω) (a) −0.5 −0.4 −0.3 −0.2 −0.1 −0.1 0.1 0.2 0.3 0.4 imag(ω) real(ω) (b)

Figure 7. Dispersion relation of singular core modes of the Lamb vortex for m = 2. (a) ℜe(ω) versus k. (b) ℜe(ω) versus ℑm(ω). The thick solid lines indicate the neutral modes obtained in section 4.1.2. The dotted lines are inviscid numerical results by Fabre et al. (2004). The Stokes line structures of the eigenfrequencies indicated by stars in figure (b) are displayed in figure 8.

0.5 1 1.5 2 2.5 3 3.5 0.5 1 1.5 2 real(r) imag(r) 0.5 1 1.5 2 2.5 3 3.5 0.5 1 1.5 2 real(r) imag(r)

(a) (b)

0.5 1 1.5 2 2.5 3 3.5 0.5 1 1.5 2 real(r) imag(r)

(c) Figure 8. Stokes line network for m = 2 modes of the Lamb vortex. (a): ω = 0.27 − 0.37i; (b): ω = 0.1285 − 0.307i; (c): ω = 0.12 − 0.315i. See caption figure 6 for explanation of the symbols.

SLIDE 20

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• S. Le Diz`

es & L. Lacaze

5 10 15 20 0.2 0.4 0.6 0.8 1 k real(ω) (a) −3 −2.5 −2 −1.5 −1 −0.5 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 1.2 imag(ω) real(ω) (b)

Figure 9. Dispersion relation of singular core modes of the Lamb vortex for m = 3. (a) ℜe(ω) versus k. (b) ℜe(ω) versus ℑm(ω). The dotted lines are inviscid numerical results by Fabre et al. (2004). The Stokes line network of the eigenfrequency indicated by a star in figure (b) is displayed in figure 10.

0.5 1 1.5 2 2.5 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 real(r) imag(r)

Figure 10. Typical Stokes line network for m ≥ 3 modes of the Lamb vortex. Here m = 3 and ω = 1.035 − 0.117i. See caption figure 6 for explanation of the symbols.

and after the bifurcation point, the network has typically the form shown on figures 8(a) and 8(c) respectively. The discontinuous behavior is therefore associated with a change

• f turning point. The dashed lines in figure 7(a,b) indicate the predictions one would

have obtained if one had kept the same turning point in formula (3.29). Asymptotically, these predictions are not good because the Stokes lines network has no longer the correct topology (as seen on figure 8(c)). Despite this point, some numerical branches are shown to follow this prediction. For finite wavenumbers, one can imagine that higher order corrections are no longer negligible and sufficiently modify the Stokes lines to jump from the configuration shown in figure 8(b), to the one shown in figure 8(a). For large wavenumbers, i. e. for large n, we are expecting all the branches to follow the solid lines

• f figures 7(a,b).

Larger values of m provide results which are all similar: frequencies are almost real near ω ≈ m − 2, and correspond to very large wavenumbers; then, they become strongly

• damped. In figure 9(a,b) the results for m = 3 are displayed. A typical Stokes line

network for one of these modes is shown in figure 10. As seen in figures 9(a,b), the numerical results do not follow the asymptotic predictions as well as for m = 1 or m = 2. This trend was already noted above for regular core modes.

SLIDE 21

An asymptotic description of vortex Kelvin modes 21

1 2 3 4 5 −4 −3 −2 −1 1 r ω +,ω −,ωc ω 1 ω 2 (a) 0.5 1 1.5 2 2.5 3 3.5 0.1 0.12 0.14 0.16 0.18 0.2 r ω +,ω −,ωc ω2 ω1 (b)

Figure 11. The functions ω± (solid lines) and ωc (dashed line) versus r for the Batchelor vortex and the parameters m = 1, kW0 = −3 (a) and m = 1, kW0 = 1.2 (b).In (a), the values ω1 and ω2 are the maximum of ωc and of ω+, respectively; In (b), they are local extrema of ω−.

To close this section on the Lamb vortex, it is important to emphasize the following

• point. We have been able to identify a few normal modes of the Lamb vortex as core
• modes. Although one can reasonably think that all the neutral modes have been iden-

tified, it is clear that an important number of inviscid damped modes do not enter the category of modes described in this section. In particular, Fabre et al. (2004) recently

• btained numerically other families of inviscid damped modes. It is possible to check by

looking at their Stokes line structure that these modes are not core modes but exhibit more complex Stokes line networks. 4.2. The Gaussian vortex with axial flow (Batchelor vortex) In this section, we attempt to account for the effect of an axial flow on the characteristics

• f the normal modes. The base flow is assumed to be given by (2.3a,b). Only neutral

modes will be considered. In particular, we shall not try to follow these modes as they become damped due to the appearance of a critical point singularity, as was done in the previous section. As already emphasized in section 3, only weak axial flow of order 1/k with k ≫ 1 can a priori be considered in the asymptotic framework. However, as the previous asymptotic results have been shown to provide good estimates for small wavenumbers without axial flow, we shall also consider here finite values of the axial flow and finite wavenumbers. As explained in section 3, the frequency intervals of regular neutral eigenmodes are

• btained by looking at the graph of the functions ω+, ω−, and ωc. Without axial flow,

we have seen that a limited number of behaviors were possible, leading to core modes

• nly. With an axial flow, others behaviors are now possible but they can only provide

ring modes. In figure 11(a,b), the functions ω+ , ω− and ωc are plotted as a function

• f the radial coordinate r, for m = 1 and kW0 = −3 and for m = 1 and kW0 = 1.2. It

can be seen on figure 11(a), that hypothesis H2 is here satisfied in the frequency interval (ω1, ω2). This means that regular neutral ring modes can be expected in this frequency

• interval. Note also that, for these parameter values, regular neutral core modes are also

expected in the frequency interval (−4, −2). In the frequency interval (ω1, ω2) shown in figure 11(b), there exist also ring modes but they are singular at the critical point which is present for large r. As for singular neutral core modes, one could show that this critical point singularity can be avoided in the complex plane without affecting the dispersion relation which can be calculated for real

• r. These modes are singular neutral ring modes. We have been able to obtain such modes

SLIDE 22

22

• S. Le Diz`

es & L. Lacaze

−5 −4 −3 −2 −1 1 2 3 −4 −2 2 4 6 kW0 ω (a) 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 −0.05 0.05 0.1 0.15 kW0 ω (b)

Figure 12. Regions of the (ω, kW0) space where neutral modes of Batchelor vortex are expected for m = 1. Pale gray, mild gray and black indicate regular neutral core mode region, singular neutral core mode region and neutral ring mode region, respectively. The large black region in (a) corresponds to regular neutral ring modes. The small singular neutral ring mode region, almost invisible in (a) is enlarged in (b). The vertical dashed lines indicated the parameter values for which ω± and ωc are plotted in figures 11(a,b) .

• nly for m = ±1. The regions of the parameter space where ring modes are expected

are indicated in black in figure 12(a) for m = 1. The region where singular neutral ring modes are expected is very small and limited to the black region shown in figure 12(b). In figures 12(a,b), the other regions in pale gray and mild gray correspond to regular neutral core modes and singular neutral core modes respectively. The region corresponding to the conditions (3.9) and (3.10) for the existence of regular neutral modes is nothing but the union of regular core mode region and regular ring mode region. This means that there is no other regular neutral modes in the white domains of figures 12(a,b). The position of ring mode and core mode regions varies with m. These variations are weak if one plots ω − m − kW0 instead of ω as demonstrated in figure 13(a-d). For negative m, the regions are obtained by the transformation (ω, k) → (−ω, −k). As seen in figure 13, neutral core modes are expected only for frequencies satisfying −2 < ω − m − kW0 < 2. By contrast, neutral ring modes are located outside this frequency interval. It is important to stress that conditions (3.9) and (3.10) exactly correspond here to regular core mode and regular ring mode regions. In particular, this implies that there is no other regular modes than those considered here. Formula (3.29) must be used in both gray regions, while formula (3.44) must be used in the black regions. As an illustration, the dispersion relation is shown in figure 14 for m = 1 and W0 = 0.3, for the first branches. Both numerical results and asymptotic predictions using (3.29) and (3.44) have been plotted. Numerical results have been

• btained by a non-viscous shooting method, with a contour deformation procedure for

the singular modes. Only neutral modes have been plotted. Numerous singular damped modes, consistent with those obtained in the previous section also exist, but they have not been displayed. Figure 14 demonstrates that formula (3.29) also works well in the case of a vortex with axial flow. Most of the branches are well approximated, except, as for the case without axial flow, the first branch, which starts from k = 0. The predictions for the regular ring modes are also fairly good. Note however that there is no singular neutral ring mode for this value of W0. Figures 15(a,b) show the radial velocity distribution of two particular eigenmodes as

• btained from the asymptotic analysis and from the numerics. A core mode is displayed

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An asymptotic description of vortex Kelvin modes 23

−8 −6 −4 −2 2 4 6 8 −4 −3 −2 −1 1 2 3 4 kW0 ω −m −kW0 (a) −8 −6 −4 −2 2 4 6 8 −4 −3 −2 −1 1 2 3 4 kW0 ω −m −kW0 (b) −8 −6 −4 −2 2 4 6 8 −4 −3 −2 −1 1 2 3 4 kW0 ω −m −kW0 (c) −8 −6 −4 −2 2 4 6 8 −4 −3 −2 −1 1 2 3 4 kW0 ω −m −kW0 (d)

Figure 13. Regions of the parameter space where neutral modes of the Batchelor vortex are

• expected. Pale gray, mild gray and black indicate regular neutral core mode region, singular

neutral core mode region and regular neutral ring mode region, respectively. (a) m = 0; (b) m = 1; (c) m = 2; (d) m = 3. The very small region obtained for m = 1 where singular neutral ring modes are expected is not indicated.

in figure 15(a). The solid curves correspond to the different leading order approximations

• btained in section 3.1 near the origin (O), in the Core region, near the turning point

(T) and in the Outer region. The thick part of each solid curve indicates the region where each approximation should apply. As expected, it is in these regions, that the asymptotic results are the closest to the numerical results (dotted curve). In figure 15(b) is shown a ring mode. In that case, we have used for the solid curves the leading order approximations obtained in section 3.2 in the Outer regions I and II, near each turning point r1 and r2 (T1 and T2), and in the Ring region. Again, a good agreement of each approximation with the numerics is obtained in the region where the approximation is expected to be valid.

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• S. Le Diz`

es & L. Lacaze

−12 −10 −8 −6 −4 −2 2 4 6 8 −3 −2 −1 1 2 3 4 k ω

−13 −12 −11 −10 −9 0.3 0.4 0.5 0.6 k ω

Figure 14. Dispersion relation of neutral modes of the Batchelor vortex for m = 1 and W0 = 0.3. Solid lines are asymptotic results. Dashed lines are numerical results. Dotted lines are the limits of the regions shown in figures 13a-d. The insert plot is a close view of the region associated with regular ring modes.

• 5. Conclusion

In this paper, an asymptotic description of the normal modes in a stable vortex has been proposed. It is based on a large axial wavenumber WKBJ analysis. An considerable effort has been made to connect the properties of the neutral modes to the characteristics

• f the base flow. In particular, we have shown how the analysis of a few quantities which

are the epicyclic frequencies ω±(r), and the critical frequency ωc(r) defined in (3.7) and (3.8), respectively, permits the regions of existence of neutral modes to be located in the parameter space. Two types of neutral normal modes have been considered, which are either confined in the vortex core (core modes) or in a ring (ring modes). General dispersion relations have been derived for both cases. In the WKBJ terminology, core modes correspond to oscillating modes between the origin and a turning point, while ring modes are oscillating modes between two distant turning points. The asymptotic dispersion relations have been applied to a Gaussian vortex with and without axial flow. For the Gaussian vortex without axial flow (Lamb vortex), neutral modes have been shown to be core modes only. Their asymptotic dispersion relation has been found to be in very good agreement with numerical results, even for small values of the wavenumber. For the Gaussian vortex with axial flow (Batchelor vortex), neutral modes have been shown to be either core modes or ring modes. A comparison with the numerics has been carried out in a single case, and a good agreement has also been observed, for both core

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An asymptotic description of vortex Kelvin modes 25

0.5 1 1.5 −0.5 −0.4 −0.3 −0.2 −0.1 0.1

r Im(u)

rt Core T Outer O (a)

0.5 1 1.5 2 2.5 3 −6 −5 −4 −3 −2 −1 1 2 3 4

r Im(u)

r2 r1 Outer I Outer II T I T II Ring (b)

Figure 15. Radial velocity profiles of eigenmodes of the Batchelor vortex for W0 = 0.3. Dotted curve: Numerical results. Solid curves: asymptotical approximations. The different regions of the asymptotical analysis are also indicated. The solid curve is thicker in the region where the approximation is expected to be valid. (a): Regular core mode (m = 1, k = 8, ω ≈ 3.91); (b) Regular ring mode (m = 1, k = −13, ω ≈ 0.27).

modes and ring modes. The spatial structure of the eigenmodes has also been shown to be well-reproduced. The influence of critical point singularities has been analyzed in detail. We have shown that core modes can remain neutral at leading order, even if they possess a critical point singularity. Such “singular neutral core modes” have been exhibited for m = 1 and m = 2 in the case without axial flow. They have also been shown to exist in presence

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−4 −3 −2 −1 1 2 3 4 −1.5 −1 −0.5 0.5 1 1.5 kW0 ω m=3 m=2 m=1 m=0 m=−1 m=−2 m=−3

Figure 16. Region in the (ω, kW0) space of possible resonance of two neutral normal modes

• f azimuthal wavenumbers m − 1 and m + 1 for various m for the Batchelor vortex.
• f axial flow. The critical point singularity damps the normal mode when it enters the

core region. We have shown that the associated complex eigenfrequencies can still be computed with the same relation applied in the complex plane by avoiding the critical point singularity in the complex plane. However, a fine monitoring of the evolution of turning points and critical points have to be performed to track the different branches. This procedure has been carried out for two families of modes for the Lamb vortex (m = 1 and m = 2). The damped core modes which are obtained in this way, have been compared favorably with recent numerical simulations by Sipp & Jacquin (2003) and Fabre et al. (2004). Interestingly, we have been able to provide an explanation to the peculiar behavior of the branches for m = 2: two different turning points have been shown to intervene in the eigenvalue relation. For the applications, it is important to emphasize that the present theory permits to

• btain information on neutral modes by a very simple procedure. We therefore expect
• ur asymptotic results to be very valuable for describing instabilities, such as the el-

liptic instability, which result from the interaction of neutral modes. As reviewed by Kerswell (2002), the elliptic instability is due to the resonance of two normal modes with the underlying strain field responsible for the elliptic deformation of the vortex. When the strain field is stationary, two neutral modes resonate if their frequencies and axial wavenumbers are identical, and their azimuthal wavenumbers differ by 2 (see e.g. Ker- swell, 2002). The present analysis allows one to easily locate the regions of the parameter space where two neutral normal modes of azimuthal wavenumbers m − 1 and m + 1 can possibly resonate; one just has to find the intersection of the regions where the two neu- tral modes exist. For the Batchelor vortex studied in section 4.2, this procedure leads to the results displayed in figure 16. In each of these regions, resonance is a priori possible. Moreover, from the nature of the mode in each region, we have the following information: resonance is possible only between regular neutral core modes and singular neutral core

• modes. In particular, this implies that ring modes cannot be resonantly excited by the

elliptical instability. The fine analysis of the resonance conditions in each region and its dependence with respect to W0 is an important issue which could have applications in

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An asymptotic description of vortex Kelvin modes 27 the aeronautical context, where the elliptical instability in a vortex with axial flow is known to be present. This will be the subject of a future study. While we have focused on stable vortices, it is worth emphasizing that the same anal- ysis could also be useful to describe unstable modes in other types of vortices. For instance, the axisymmetric unstable modes associated with the centrifugal instability can be captured by our analysis. These modes are stationary and localized in the radial region where the Rayleigh discriminant Υ(r) = 2Ω(r)ζ(r) is negative (e.g. Rossi, 2000). Bayly (1988) demonstrated that a large k-asymptotic analysis could be possible to de- scribe these modes. With our terminology, these modes would be ring modes localized between two turning points satisfying ∆ = 0, i.e. Φ(r) = −ℑm(ω). The most unstable modes would correspond to the configuration where the two turning points are close to the minimum of Υ. We refer to Bayly (1988) and Rossi (2000) for details. Gallaire & Billant (2003) recently showed that the same analysis could also be used to extend the Rayleigh instability criterion to non-axisymmetric modes. This work has benefited from discussions with D. Fabre, A. Antkowiak and F. Gallaire. We would also like to thank D. Sipp and D. Fabre for having provided the numerical data plotted in figures 5, 7 and 9. Thanks also to Kris Ryan for his careful reading of the manuscript.

References

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. New York: Dover. Arendt, S., Fritts, D. C. & Andreassen, O. 1997 The initial value problem for Kelvin vortex waves. J. Fluid Mech. 344, 181–212. Ash, R. L. & Khorrami, M. R. 1995 Vortex stability. In Fluid Vortices (ed. S. I. Green), chap. VIII, pp. 317–372. Kluwer Academic Publishers. Bayly, B. J. 1988 Three-dimensional centrifugal-type instabilities in inviscid two- dimensional flows. Phys. Fluids 31 (1), 56–64. Bender, C. M. & Orszag, S. A. 1978 Advanced mathematical methods for scientists and engineers. New York: McGraw-Hill. Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic stability. Cambridge University Press. Fabre, D. 2002 Instabilitit´ es et instationnarit´ es dans les tourbillons: application aux sillages d’avions. PhD thesis, ONERA/Universit´ e Paris VI. Fabre, D., Sipp, D. & Jacquin, L. 2004 The Kelvin waves of the Lamb-Oseen vortex.

• J. Fluid Mech. (submitted).

Fedoryuk, M. V. 1993 Asymptotic analysis. Springer verlag. Gallaire, F. & Billant, P. 2003 Generalized Rayleigh criterion for non-axisymmetric centrifugal instabilities. Bull. Am. Phys. Soc. 48 (10), 66. Greenspan, H. P. 1968 The theory of rotating fluids. Cambridge University Press. Kerswell, R. R. 2002 Elliptical instability. Ann. Rev. Fluid Mech. 34, 83–113. Landau, L. & Lifchitz, E. 1966 M´ ecanique Quantique, Th´ eorie non relatiste. ´ Editions MIR. Le Diz` es, S. 2004 Viscous critical-layer analysis of vortex normal modes. Stud. Appl.

• Math. 112, 315–332.

Lin, C. C. 1955 The theory of hydrodynamics stability. Cambridge University Press. Olver, F. W. J. 1997 Asymptotics and special functions. A K Peters, Ltd.

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28

• S. Le Diz`

es & L. Lacaze Rossi, M. 2000 Of vortices and vortical layers: an overview. In Vortex Structure and Dynamics (ed. A. Maurel & P. Petitjeans), pp. 40–123. Springer Verlag. Saffman, P. G. 1992 Vortex dynamics. Cambridge University Press. Sipp, D. & Jacquin, L. 2003 Widnall instabilities in vortex pairs. Phys. Fluids 15 (7), 1861–1874. Stewartson, K. & Brown, S. 1985 Near-neutral-centre-modes as inviscid perturba- tions to a trailing line vortex. J. Fluid Mech. 156, 387–399. Stewartson, K. & Leibovich, S. 1987 On the stability of a columnar vortex to disturbances with large azimuthal wavenumbers: the upper neutral points. J. Fluid

• Mech. 178, 549–566.